This cross-disciplinary project develops new methods and explicates the theory and applications of new techniques for statistical inference for quantum systems. Many domains of basic biomedical research are now near or will soon be in quantum-noise limited regimes. Familiar, classical statistical methods are not applicable for such data because, basically, the usual rules for sampling, distributions and inference are no longer valid. In place of the classical methods of statistical inference and biomedical data analysis, mathematically rigorous alternatives developed by Helstrom, Holevo and many others, in the last twenty years are used. The project has included extending the basic theory, finding connections with Jordan algebras, and studying a range of plausible applications for significant biomedical problems in detection and estimation.